By law, as a U.S. Citizen (even with dual citizenship), you are required to file a return every year claiming your world wide income.
The U..S. allows a generous foreign income exclusion, but to get it, you have to apply for it and file the return.
You are not alone in this short coming. Thousands of American living abroad are not meeting their filing requirements.
The tax treaty, allows the country in which you are living and earning income to tax your foreign earned income first. The use then taxes the amount that is in excess of the foreign earned income exclusion, and gives you credit for taxes paid to the foreign country, in this case, Ireland.
Your tax obligation to the U.S. is that the stock that is purchases from the U.S. exchanges is taxed first by the U.S. when you sell them as short term or long term gains.
You would figure any short or long term gains and losses on schedule D.
If there is a loss, then no tax is due because there is no capital gain. The capital loss reduces your taxable income.
The W-9 is filled out by you indicating if you are subject to back up with holding or not. Since you are a citizen living abroad (non-resident) most likley you would be subject to backup withholding.
The W-9 is not sent to the IRS, contrary to popular belief. It is maintained on file by the business or in this case, the brokerage as part of their records to substantiate that you disclosed the ssn and that you indicated if you were subject to backup withholding or not. They would use it to protect themselves in an audit.
FYI: your UK based broker does not have a reporting requirement to the IRS.